Thanks for your question. You are correct in two ways: (1) according to
the theory of relativity, all observers will measure the same speed of
light, and (2) our Galaxy and other galaxies are so large that light takes
a long time to cross the width of a galaxy.

It does not follow, however, that there should be significant distortion in
a galaxy from this effect. This is because the motions of galaxies
and the motions of objects within galaxies are very slow compared to
the speed of light.

Consider the situation in our Milky Way, and assume (as you do) that our
Galaxy is typical of other systems. Imagine looking at our Galaxy from far
off, and consider two stars that are orbiting around as the Sun does at a
distance of about 25,000 light years from the center of the Galaxy.
Imagine two stars, one on the near side and the other on the far side.

Now imagine that we take an observation with a telescope. The light from
the star on the far side was emitted 50,000 years earlier than the
light from the star on the near side. The question to ask is how far that
star has moved during those 50,000 years. The distortion you are
postulating will be equal to the angle (in degrees) that the star will move
in its orbit during the light travel time across the Galaxy. This is
equivalent to asking where the far-side star was 50,000 years
before the moment its light passed the near-side star (though this is quite
non-relativistic thinking!).

Our Sun takes about 250 million years to orbit around the Galaxy. In
50,000 years, it travels a certain fraction of the way around its orbit,
namely

360 degrees * [(50,000 years) / (250,000,000 years)],

or about 0.07 degree, equivalent to 0.0002 of its orbit. So the distortion
would be only about 0.0002 of the image size, which is quite small indeed!

You would get similar answers if you ask about the distortion of galaxies
as they move through space. This motion is only a few hundred km/sec,
again very much smaller than the speed of light.

In general, relativistic effects are only important for objects moving at a
large fraction of the speed of light.